London-Paris Bachelier Workshop

Funding inclusive valuation as modified option
pricing

Cristin Buescu - King’s College London—Joint work with Damiano Brigo and Marek Rutkowski

30th September, 2016
1 / 24

The Idea
Let F = (Ft ) where Ft := (Su , u  t) for a traded asset (stock).

For fixed maturity T let X be an FT -measurable integrable rv. andlet A be a contract (vulnerable call option) that costs P0 at time 0
and has the payoff X at maturity time T
X = 1{⌧ >T } (ST K )+with default time ⌧ being a positive rv on (⌦, F, P).
The price Pt , t 2 [0, T ], of this contract in a market withi) funding costs: unsecured funding account with the interest rate f ;ii) stock (the underlying asset of the contract);iii) repo agreement on the stock with the repo rate h (at most %);iv) credit risk: zero-recovery defaultable bond with the rate of return r C(one-to-one with CDS).

is of Black-Scholes w. dividends type (modified option pricing)) sensitivity analysis.
2 / 24Agenda

1

Historical perspective

2Pricing benchmark products with funding, repo and credit
Modified cash-flow approach: Black-Scholes w. dividends
Martingale method approach: Black-Scholes w. dividends

3

Sensitivity analysis via the "funding Greeks"

4Conclusions

3 / 24

Historical perspective

Timeline:credit risk not considered pre-crisis
2008: 8 major credit events in 1 month (Sep 7 to Oct 8)
sudden divergence between OIS and LIBOR(sign of presence of credit and liquidity risk affecting notional)
impact on valuation of contracts: adjustments
Adjustments
CVA (counterparty default cost)
DVA (own default benefit)
benefit goes to creditors, not shareholders; discard it? (Albanese,Caenazzo, Crepey 2016)
can book large profit (Citigroup $2.5 bn in 2009)difficult to hedge (via correlated proxies)

FVA (cost of funding the trade: treasury, creditors)
large! ($1.5 bn JPM in 2014)

KVA (cost of capital used)no commonly agreed definition
not aware of amounts yet
4 / 24

Historical perspective

Context
These contracts are OTC
adjustments can provide a starting negotiating position
move them to CCP? (challenge: reduce systemic risk while
keeping trades feasible)
market size still very large: $24.7 tn of derivatives (2012), $632 tn
notional (BIS 2013)
generally move towards simpler standardized contracts, shying
away from features that may blow up
Issue
Valuation becomes a nonlinear recursive problem
Solution involves semi-linear PDEs and BSDEs (see the works ofEl Karoui, Peng, Quenez or Crepey)
Get ad-hoc numerical solutions that are time-consuming anddifficult to work with
5 / 24

Historical perspective

Goal: "Our life is frittered away by detail... simplify, simplify."
Henry David Thoreau

standard benchmark product (vulnerable call option)account for adjustments in price by including corresponding costs
and benefits in the cash flow (approach of Brigo, Pallavicini 2014)
derive and solve pricing equation via modified option pricing
reconcile it with martingale measure approach (Bielecki,Jeanblanc, Rutkowski 2005)
use it for sensitivity analysis: funding rate, repo, credit spread
6 / 24

Pricing benchmark products with funding, repo and credit

Modified cash-flow approach: Black-Scholes w. dividends

Adjusted cash flow approach (Brigo, Pallavicini 2014)
Their pricing equation (11) becomes for our benchmark product⇥⇤Vt = E h 1{⌧ >T } D(t, T ; f )(ST K )+ | Gt

(1)

where Qh is s.t. the drift of the risky asset is h:dSt = hSt dt + St dWth
G = (Gt ) is the full filtration including default information and thediscount factor is✓ Z t◆
D(s, t; f ) := exp
fu du .
s

Assume a constant treasury rate f and use the counterparty
pre-default intensity under Qh defined in their equation (40)1{⌧ >t} dt := Qh (⌧ 2 dt | ⌧ > t, Ft ),to obtain the survival probability Gth := Qh (⌧ > t | Ft ) = et.
7 / 24

Pricing benchmark products with funding, repo and credit

Modified cash-flow approach: Black-Scholes w. dividends

Change of filtration formula (Cor. 3.1.1 in Bielecki, Jeanblanc,Rutkowski 2004) gives:
Vt = 1{⌧ >t} (Gth ) 1 Eh [D(t, T ; f )(STK )+ GTh | Ft ].

e denotes the F-adapted pre-default price process s.t. 8t 2 [0, T ]If Vet ,1{⌧ >t} Vt = 1{⌧ >t} VthenetV= e ( +f )(Tt)

Eh [(ST= e ( +fh)(T t)= e ( +fh)(T t)

K )+ | Ft ]Eh [e h(TBlack-Scholes with dividend ratet)

St N(d1 )+f(STK )+ | Ft ]Ke h(Tt)

N(d2 ) .

h and discount

(2)

+ f.
8 / 24

Pricing benchmark products with funding, repo and credit

Modified cash-flow approach: Black-Scholes w. dividends

Main result (adjusted cash flow)
The valuation of a zero-recovery vulnerable call option in the presenceof funding costs and repo can be mapped to the Black-Scholes formulawith dividends⇣⌘
Vt = 1{⌧ >t} St e ( +f h)(T t) N(d1 ) Ke ( +f )(T t) N(d2 ) . (3)

9 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Martingale method approach
Replication
Let F = (Ft ) where Ft := (Su , u  t) for a traded asset (stock).
For fixed maturity date T let X be an FT -measurable integrable
random variable.

Assume that the default time ⌧ is a positive random variable on
the probability space (⌦, F, P). It generates a filtration H = (Ht )where Ht := (1{⌧ u} , u  t), which is used to progressively
enlarge F in order to obtain the full filtration G = (Gt ) with
Gt := Ft _ Ht .

Assume that Ft := P(⌧  t | Ft ) is a continuous, increasing
function and Ft < 1 for any t (see Elliott, Jeanblanc, Yor (2000) inconjunction with the hypothesis (H) of Kusuoka)

10 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Replication of a defaultable bond using CDS
Motivation for the assumption on the distribution of ⌧Replicate a zero-recovery defaultable bond maturing at T with afunding account ( ft ) and a CDS on the bond issuer (spread r CDS ).
The price process B in terms of the point process J (jumps to onee isat default) and of the pre-default price B
e t = 1{⌧ >t} B
etBt = 1{Jt =0} B

Assume that if no default occurred before t, then between t andt + dt default may happen with a positive probability for arbitrarily
small dt.

11 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

At time t < ⌧ ^ T :e t from the treasury and buy one defaultable bond;1borrow Be t of CDS contracts on the same name.
2
buy a number B
At time t + dt:
e t CDS3
if there is a default (i.e., Jt+dt = 1), then each of the B
contracts pays 1;
e t+dt ;4
if there is no default (i.e., Jt+dt = 0), then sell the bond for B
e t CDS
5either way, pay the premium leg r CDS dt for each of the B
econtracts and pay back the loan to the treasury: Bt (1 + ft dt).
The overall gain over the time interval (t, t + dt) is
e t r CDS dt
Be t 1{J =1} + B
e t+dt 1{J =0}
Bt+dtt+dt

e t (1 + ft dt).
B

Equating this to zero to ensure replication gives dynamics:
dBt

Bt (r CDS + ft ) dt + Bt dJt = 0,(4)and thus, since BT = 1{⌧ >T } , for all t 2 [0, T ]Bt = 1{⌧ >t} eRTt(r CDS +fu ) du.

(5)
12 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Proposition
The above replication of the defaultable bond holds whenever theprobability distribution of ⌧ is continuous and its support includes [0, T ].
This motivates our assumption on the distribution of the default
time, and renders the replication independent of the particular
distribution in this class.
The replication should not be postulated a priori without thisassumption.
For instance, it fails when P(⌧ 2 (t1 , t2 )) = 0 (see the analysis ofRutkowski (1999) for the discontinuous case).

13 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Alternative derivation of bond pricing
The replication can also be derived by martingale methods, taking
any measure Q equivalent to P as postulated martingale measure.
Key step: for any positive density on [0,T] 9! measure Q
equivalent to P s.t. the distribution of ⌧ under Q is exp( )It can be shown that Q is unique on Ht (information on defaults),
so the model is complete and FTAP yields:
Bt = 1{⌧ >t} e (+f )(Tt)(it can be shown that  equals the CDS spread r CDS )14 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Let A be a contract (vulnerable call option) that costs P0 at time 0 andhas the payoff X at maturity time T
X = 1{⌧ >T } (STK )+ .

Want the price Pt , t 2 [0, T ], of this contract for an investor who
replicates a long position using available financial instruments.
The market has primary assets (A1 , A2 , A3 , A4 ):i) unsecured funding account with the interest rate f ;ii) stock (the underlying asset of the contract);iii) repo agreement on the stock with the repo rate h;iv) zero-recovery defaultable bond with the rate of return r C issued by
the counterparty.
(Recall one-to-one correspondence between r C and r CDS if needed.)At time t, the price Pti of the asset Ai is given by
Pt1 = Btf , Pt2 = St , Pt3 = 0, Pt4 = Bt
15 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

A trading strategy ' = ('1 , '2 , '3 , '4 ) gives the number of units ofeach primary asset purchased to build a portfolio.
A trading strategy ' is admissible if the repo is used for a fraction
the required amount of stock, and the rest is funded by treasury.
At time t 2 [0, T ] the wealth corresponding to an admissible ' isVt' =4
Xof'it Ptii=1

A strategy ' is self-financing if for all t 2 [0, T ]Vt' = V0' + Gt' .

(6)

An admissible trading strategy ' replicates the payoff of a contract A if
VT' = X .
The time t price of a contract A is the wealth Vt'
Pt := Vt' .

(7)

The existence of the specific primary assets in our market ensures thatany claim is attainable.
16 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

The replicating strategy replicates not only the payoff of the option, butalso the credit risk profile of a long position in the option.
The investor
1buys
t repos, borrows
t shares, and receives
treasury;t St from treasury to buy and delivert St cash which is paid back to

2borrows (13buys Pt /Bt units of the counterparty bond in order to match thevalue of this portfolio and the option payoff.

)t St from treasury and buys (1)t shares;This portfolio corresponds to the following the admissible strategy!
(1) t StPt✓t :=, (1) t,.
t,Bt
Btf

(8)

17 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Standard replicating condition, self-financing and Ito lead to thepre-default pricing PDE for the function v (t, s)@v+@s

2 s 2 @v 2rtC v = 0(9)The time t price of the vulnerable call option equals⇣⌘C
Pt = 1{⌧ >t} St e q(T t) N(d1q ) Ke r (T t) N(d2q )(10)

vt + (1)f + h s

with terminal condition v (T , s) = (s2 @s2K )+ .

Main result (replication)with q = r C f and the effective funding rate defined as theweighted average: f := (1)f + h.
For= 1 (repo only) this is the same as for adjusted cash flow.
18 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Martingale method

The same pricing mapping into Black-Scholes w. dividends can bederived without resorting to PDEs
Once we have the wealth dynamics use Q the probability
measure equivalent to P s.t. the drift of St under Q is the effective
funding rate f
This yields a probabilistic representation of the price as a
discounted expectation of the payoff under Q and Black-Scholes
formula again

19 / 24

Pricing benchmark products with funding, repo and credit

Martingale method approach: Black-Scholes w. dividends

Consistency

Valuation of contracts under funding, repo and credit byadjusted cash flow approachmartingale measure approachmaps into Black-Scholes formula w. dividends.
Explicit formula allows for sensitivity analysis⇣CPt = 1{⌧ >t} St e q(T t) N(d1q ) Ke r (Twith q = r Cf , rC =+ f , f := (1)f + h,t)N(d2q )⌘= r CDS .

Q: Which has the most price impact: funding, repo, or credit?

20 / 24Sensitivity analysis via the "funding Greeks"

Numerical example
For St = 80, K = 100, = 0.3, T t = 0.1, r CDS = 0.05 the pre-default
price of the vulnerable call is decreasing in the funding rate fincreasing in the repo rate h.

21 / 24Sensitivity analysis via the "funding Greeks"Introduce and evaluate "funding Greeks":@f< 0 (f treasury rate)@h > 0 (h repo rate)@> 0(is CDS spread)Compute relative sensitivities:
@h VV
@f VV
@ VV> Tt

(repo)=(Tt)(funding)=(Tt)(credit)

Valuation impact: repo > funding = credit.
22 / 24Sensitivity analysis via the "funding Greeks"

Intepretation of "funding Greeks"vulnerable call = hybrid product: call on stock + long on bondif repo is used (0 <  1) the price increases in h (repo rate) due
to the cost of hedging the option
dual impact of f :borrow cash (V increases with f )invest in bond with rate f + r CDS (V decreases with f )overall impact of f may be negative as in the example (clear in thecase = 1 with no borrowing)

23 / 24

Conclusions

Conclusions
Funding is not just a spread, but a complex nonlinear andrecursive pricing problem (see expression below to appreciate
what our simplification achieved)
Pt=Z Tt+h

E [1{u<⌧ } D(t, u; f )(⇧(u, u + du) + 1{⌧ 2du} ✓u )|Gt ]Z TthE [1{u<⌧ } D(t, u; f )((fuNCcu )Mu + (fuCNIcu )Nu + (fuIcu )Nu )|Gt ]duTwo alternative pricing approaches lead to the same result for abenchmark product when including funding, credit and repoValuation (with adjustments) is mapped into Black-Scholes
formula with dividends
This allows for sensitivity analysis
Pricing impact of repo rate is larger than that of funding or credit
24 / 24

Conclusions

References

References
Albanese, C., Caenazzo, and Crepey, S. (2016) Capital valuation adjustment and funding valuation adjustment. Working
paper available at http://arxiv.org/abs/1603.03012
Bielecki, T. R., Jeanblanc, M. and Rutkowski, M. (2004) Modeling and valuation of credit risk. In Stochastic Methods inFinance, edited by M. Frittelli and W. Runggaldier, Springer: Berlin, pp. 27–126.
Bielecki, T. R., Jeanblanc, M. and Rutkowski, M. (2005) PDE approach to valuation and hedging of credit derivatives.
Quantitative Finance, 5 (3): 257–270.
Bielecki, T. R., Jeanblanc, M. and Rutkowski, M. (2009) Credit Risk Modeling. (Osaka University Press).
Brigo, D., Buescu, C. and Rutkowski, M. (2016) Funding, repo and credit inclusion in option pricing via dividends Capitalvaluation adjustment and funding valuation adjustment. Working paper available at
http://arxiv.org/abs/1602.05998
Brigo, D. and Pallavicini, A. (2014) Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial
margins under credit, funding and wrong-way risks. Journal of Financial Engineering, 1 (1): 1–60.
El Karoui, N., Peng, S. and Quenez, M. C. (1997) Backward Stochastic Differential Equations in Finance. MathematicalFinance, 7(1): 1–71.
Elliott, R.J., Jeanblanc, M. and Yor, M. (2000) On models of default risk. Mathematical Finance, 10 (2): 179–195.
Rutkowski, M. (1999) “On Models of Default Risk" by R. Elliott, M. Jeanblanc and M. Yor. Working paper.

25 / 24